The u and v coordinates
are the direction cosines of a vector with respect to the y-axis
and z-axis, respectively.
The u/v coordinates for
the hemisphere x ≥ 0 are derived from the phi and theta angles,
as follows:
$$\begin{array}{l}u=\mathrm{sin}\theta \mathrm{cos}\varphi \\ v=\mathrm{sin}\theta \mathrm{sin}\varphi \end{array}$$
In these expressions, φ and θ are the phi and theta
angles, respectively.
In terms of azimuth and elevation, the u and v coordinates
are
$$\begin{array}{l}u=\mathrm{cos}el\mathrm{sin}az\\ v=\mathrm{sin}el\end{array}$$
The values of u and v satisfy
the inequalities
$$\begin{array}{l}-1\le u\le 1\\ -1\le v\le 1\\ {u}^{2}+{v}^{2}\le 1\end{array}$$
Conversely, the phi and theta angles can be written in terms
of u and v using
$$\begin{array}{l}\mathrm{tan}\varphi =u/v\\ \mathrm{sin}\theta =\sqrt{{u}^{2}+{v}^{2}}\end{array}$$
The azimuth and elevation angles can also be written in terms
of u and v
$$\begin{array}{l}\mathrm{sin}el=v\\ \mathrm{tan}az=\frac{u}{\sqrt{1-{u}^{2}-{v}^{2}}}\end{array}$$
The φ angle is the angle from the positive y-axis
toward the positive z-axis, to the vector's
orthogonal projection onto the yz plane. The φ
angle is between 0 and 360 degrees. The θ angle is the angle
from the x-axis toward the yz plane,
to the vector itself. The θ angle is between 0 and 180 degrees.
The figure illustrates φ and θ for a vector that
appears as a green solid line. The coordinate system is relative to
the center of a uniform linear array, whose elements appear as blue
circles.
The coordinate transformations between φ/θ and az/el are
described by the following equations
$$\begin{array}{l}\mathrm{sin}(\text{el})=\mathrm{sin}\varphi \mathrm{sin}\theta \hfill \\ \mathrm{tan}(\text{az})=\mathrm{cos}\varphi \mathrm{tan}\theta \hfill \\ \hfill \\ \mathrm{cos}\theta =\mathrm{cos}(\text{el})\mathrm{cos}(\text{az})\hfill \\ \mathrm{tan}\varphi =\mathrm{tan}(\text{el})/\mathrm{sin}(\text{az})\hfill \end{array}$$
The azimuth angle is
the angle from the positive x-axis toward the positive y-axis,
to the vector's orthogonal projection onto the xy plane.
The azimuth angle is between –180 and 180 degrees. The elevation
angle is the angle from the vector's orthogonal
projection onto the xy plane toward the positive z-axis,
to the vector. The elevation angle is between –90 and 90 degrees.
These definitions assume the boresight direction is the positive x-axis.
Note:
The elevation angle is sometimes defined in the literature as
the angle a vector makes with the positive z-axis.
The MATLAB^{®} and Phased Array System Toolbox™ products do not
use this definition. |
This figure illustrates the azimuth angle and elevation angle
for a vector that appears as a green solid line. The coordinate system
is relative to the center of a uniform linear array, whose elements
appear as blue circles.